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This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.

We prove a localized non blow-up theorem of the Beale–Kato–Majda type for the solution of the 3D incompressible Euler equations.

In this paper we prove local in time well-posedness for the incompressible Euler equations in R n for the initial data in L 1 ( 1 ) 1 ( R n ) , which corresponds to a critical case of the generalized Campanato spaces L q ( N ) s ( R n ) . The space is studied extensively in our companion paper [9] , and in the critical case we have embeddings B ∞ , 1 1 ( R n ) ↪ L 1 ( 1 ) 1 ( R n ) ↪ C 0 , 1 ( R n ) , where B ∞ , 1 1 ( R n ) and C 0 , 1 ( R n ) are the Besov space and the Lipschitz space respectively. In particular L 1 ( 1 ) 1 ( R n ) contains non- C 1 ( R n ) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L 1 ( 1 ) 1 ( R n ) , for which the solution to the Euler equations blows up in finite time.

We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces, B p , q s , where the parameters p , q and s considered here are such that the local existence and uniqueness result holds.

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.

In this paper, we study the desingularization of multiscale rotating vortex patches with fixed angular velocity to the 2D Euler equations in a disk. We prove the existence of two-parameter multiscale V-states concentrating near two points. These two points are determined by the Robin function to the Green’s function, the Green’s function and angular velocity, and not symmetric about the origin, which is new to the former results. The existence of multiscale V-states with N-folds for $$ N\ge 2 $$ is also proved.

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of traveling vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal intensities but opposite signs. The results are obtained by using an improved vorticity method.

We prove the global well-posedness of the free interface problem for the two-phase incompressible Euler Equations with damping for the small initial data, where the effect of surface tension is included on the free surfaces. Moreover, the solution decays exponentially to the equilibrium.

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $$L^2$$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.

By using an idea of localized Galilean boost, we show that the data-to-solution map for incompressible Euler equations is not uniformly continuous in $${H^s({\mathbb{R}}^d)}$$, $${s \ge 0}$$. This settles the end-point case (s = 0) left open in Himonas–Misiolek (Commun Math Phys 296(1):285–301, 2010) and gives a unified treatment for all Hs. We also show the solution map is nowhere uniformly continuous.

We consider the initial value problem for the Nernst-Planck equations coupled to the incompressible Euler equations in $\mathbb T^2$. We prove global existence of weak solutions for vorticity in $L^p$. We also obtain global existence and uniqueness of smooth solutions. We show that smooth solutions of the Nernst-Planck-Navier-Stokes equations converge to solutions of the Nernst-Planck-Euler equations as viscosity tends to zero. All the results hold for large data.

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question, by first describing a class of potentially singular solutions to the Euler equations numerically discovered in axisymmetric geometries, and then by presenting evidence from rigorous analysis that strongly supports the existence of such singular solutions. The initial data leading to these singular solutions possess certain special symmetry and monotonicity properties, and the subsequent flows are assumed to satisfy a periodic boundary condition along the axial direction and a no-flow, free-slip boundary condition on the solid wall. The numerical study employs a hybrid 6th-order Galerkin/finite difference discretization of the governing equations in space and a 4th-order Runge--Kutta discretization in time, where the emerging singularity is captured on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10¹²)² near the point of the singularity, the simulations are able to advance the solution to a point that is asymptotically close to the predicted singularity time, while achieving a pointwise relative error of O(10⁻⁴) in the vorticity vector and obtaining a 3 x 10⁸-fold increase in the maximum vorticity. The numerical data are checked against all major blowup/nonblowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A close scrutiny of the data near the point of the singularity also reveals a self-similar structure in the blowup, as well as a one-dimensional model which is seen to capture the essential features of the singular solutions along the solid wall, and for which existence of finite-time singularities can be established rigorously.

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfies an elliptic equation on each of the hypersurfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation., Comment: 7 pages. Version 2: Improved wording and presentation

The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained.

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H 3 , with the H 3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.

The quasineutral limit of the isothermal Navier–Stokes–Poisson system is rigorously proved when the combined quasineutral and vanishing viscosity limit is considered in a domain with boundary. The convergence of the global weak solution for Navier–Stokes–Poisson system to the strong solution for incompressible Euler equations is obtained.

We consider the Cauchy problem for the incompressible Euler equations on R d for d ≥ 3 . Then we demonstrate the local-in-time solvability of classical solutions with the nondecaying derivatives and finite kinetic energy. Moreover, we establish the blowup criterion of such solutions in terms of the vorticity.

We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes., 14 pages

In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions. We establish the existence of subsequences conv...

In [Grenier-Nguyen], we introduced so called {\em generators} functions to precisely follow the regularity of analytic solutions of Navier Stokes equations. In this short note, we give a presentation of these generator functions and use them to give existence results of analytic solutions to some classical equations, namely to hyperbolic equations, to incompressible Euler equations, and to hydrostatic Euler and Vlasov models. The use of these generator functions appear to be an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem [Asano,Caflisch,Nirenberg,Safonov]., 12 pages

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

We study weak solutions of the incompressible Euler equations on T2×R+; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that u∈L3(0,T;L3(T2×R+)), lim|y|→01|y|∫0T∫T2∫x3>|y|∞|u(x+y)−u(x)|3dxdt=0, and an additional continuity condition near the boundary: for some δ>0 we require u∈L3(0,T;C0(T2×[0,δ])). We note that all our conditions are satisfied whenever u(x,t)∈Cα, for some α>1/3, with Holder constant C(x,t)∈L3(T2×R+×(0,T)).

We prove a continuation critereon for incompressible liquids with free surface boundary. We combine the energy estimates of Christodoulou and Lindblad with an analog of the estimate due to Beale, Kato, and Majda for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary remain bounded, assuming that the Taylor sign condition holds., Corrected typos

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem., 7 pages, 5 figures, 1 supplemental video

This paper mainly concerns the mathematical justification of the asymptotic limit of the Gross-Pitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.

We show that the 2D Euler equations are not locally well-posed in the sense of Hadamard in the \(C^1\) space and in the Besov space \(B^1_{\infty ,1}\). Our approach relies on the technique of Lagrangian deformations of Bourgain and Li (Strong ill-posedness of the incompressible Euler equations in borderline Sobolev spaces. arXiv:1307.7090). We show that the assumption that the data-to-solution map is continuous in either \(C^1\) or \(B^1_{\infty ,1}\) leads to a contradiction with a result in \(W^{1,p}\) of Kato and Ponce (Rev Mat Iberoam 2:73–88, 1986).

This paper investigates temporal regularity of solutions for the incompressible Euler equations in a critical Besov space B d p +1 p,1 (R ) for 1 ≤ p ≤ d.

International audience; We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold's interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.

In this work we investigate the helicity regularity for weak solutions of the incompressible Euler equations. To prove regularity and conservation of the helicity we will treat the velocity u u and its curl ⁡ u \operatorname {curl} u as two independent functions and we mainly show that the helicity is a constant of motion assuming u ∈ L t 2 r ( C x θ ) u \in L^{2r}_t(C^\theta _x) and curl ⁡ u ∈ L t κ ( W x α , 1 ) \operatorname {curl} u \in L^{\kappa }_t(W^{\alpha ,1}_x) , where r , κ r,\kappa are conjugate Hölder exponents and 2 θ + α ≥ 1 2\theta +\alpha \geq 1 . Using the same techniques we also show that the helicity has a suitable Hölder regularity even in the range where it is not necessarily constant.

In this paper we consider the locally backward self-similar solutions for the Euler system, and focus on the case that the possible nontrivial velocity profiles have non-decaying asymptotics. We derive the meaningful representation formula of the pressure profile in terms of velocity profiles in this case, and by using it and the local energy inequality of profiles, we prove some nonexistence results and show the energy behavior concerning the possible velocity profiles., Comment: 18 pages

Starting from Brenier’s relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn’s algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem (see Arnaudon et al. in An entropic interpolation problem for incompressible viscid fluids, 2017, arXiv preprint arXiv:1704.02126 ). Numerical results in dimension one and two illustrate the feasibility of the method.

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty} (B(x_0, r)))$ of the 3D Euler equations, where $B(x_0,r)$ is the ball with radius $r$ and the center at $x_0$, if the limiting values of certain scale invariant quantities for a solution $v(\cdot, t)$ as $t\to 0$ are small enough, then $ \nabla v(\cdot,t) $ does not blow-up at $t=0$ in $B(x_0, r)$., 22 pages

In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |\Delta v|^{\frac65} dx, Comment: 15 pages(to appear in Comm. Math. Phys.). arXiv admin note: substantial text overlap with arXiv:1105.3639

For any \theta, Comment: 40 pages

We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.

International audience; In this paper we propose a large set of analytical solutions (FRESH-ASSESS) for the hydrostatic incompressible Euler system in 2d and 3d. These solutions mainly concern free surface flows but partially free surface flows are also considered. These analytical solutions can be especially useful for the validation of numerical schemes.

In 2000 Constantin showed that the incompressible Euler equations can be written in an "Eulerian-Lagrangian" form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain H��lder spaces $C^{1,��}$. We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in $H^s$ for $n\geq2$ and $s>\frac{n}{2}+1$, a unique local-in-time solution exists on the $n$-torus that is continuous into $H^s$ and $C^1$ into $H^{s-1}$. These solutions automatically have $C^1$ trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory., 17 pages, to appear in J. Math. Fluid Mech. Lemmas 4 and 6 revised, several minor changes

In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming a small loop provided that they are close enough. This confirms the numerical observations done in the recent work [10], 32 pages

We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions., AMS Latex, 15 pages

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits., Comment: 81 pages

In this paper we study a generalization of self-similar solutions. We show that just as for the solutions to the Navier–Stokes equations these supposedly singular solution reduce to the zero solution.

In this paper, local well-posedness for the density dependent incompressible Euler equations is established in Besov spaces. We also obtain a blow-up criterion for the corresponding solution.

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with smooth boundary. A class of additional boundary conditions for the vorticities are identified so that the solution is unique and stable.